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TAM 212: Introductory Dynamics

Calculus and vectors #rvc

Time-dependent vectors can be differentiated in exactly the same way that we differentiate scalar functions. For a time-dependent vector a(t), the derivative ˙a(t) is:

Vector derivative definition.#rvc‑ed

˙a(t)=ddta(t)=limΔt0a(t+Δt)a(t)Δt

Note that vector derivatives are a purely geometric concept. They don't rely on any basis or coordinates, but are just defined in terms of the physical actions of adding and scaling vectors.

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Increment: Δt= 2 s
Time: t= 0 s

Vector derivatives shown as functions of t and Δt. We can hold t fixed and vary Δt to see how the approximate derivative Δa/Δt approaches ˙a. Alternatively, we can hold Δt fixed and vary t to see how the approximation changes depending on how a is changing. #rvc‑fd

We will use either the dot notation ˙a(t) or the full derivative notation da(t)dt, depending on which is clearer and more convenient. We will often not write the time dependency explicitly, so we might write just ˙a or dadt.

Derivatives and vector “positions” #rvc‑sp

When thinking about vector derivatives, it is important to remember that vectors don't have positions. Even if a vector is drawn moving about, this is irrelevant for the derivative. Only changes to length and direction are important.

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Movement: bounce stretch circle twist slider rotate vertical fly

Vector derivatives for moving vectors. Vector movement is irrelevant when computing vector derivatives. #rvc‑fp

Derivatives in components #rvc‑sc

In a fixed basis we differentiate a vector by differentiating each component:

Vector derivative in components#rvc‑ec

˙a(t)=˙a1(t)ˆı+˙a2(t)ˆȷ+˙a3(t)ˆk

Derivation

Warning: Differentiating each component is only valid if the basis is fixed. #rvc‑wc

Time: t= 0 s
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Basis: ˆı,ˆȷ ˆu,ˆv

The vector derivative decomposed into components. This demonstrates graphically that each component of a vector in a particular basis is simply a scalar function, and the corresponding derivative component is the regular scalar derivative. #rvc‑fc

Differentiating vector expressions #rvc‑se

We can also differentiate complex vector expressions, using the sum and product rules. For vectors, the product rule applies to both the dot and cross products:

Product rule for dot-product derivatives.#rvc‑ep

ddt(ab)=˙ab+a˙b

Product rule for cross-product derivatives.#rvc‑ex

ddt(a×b)=˙a×b+a×˙b

Example Problem: Differentiating vector product expressions. #rvc‑xe

The chain rule also applies to vector functions. This is helpful for parameterizing vectors in terms of arc-length s or other quantities different than time t.

Chain rule for vectors.#rvc‑er

ddta(s(t))=dads(s(t))dsdt(t)=dads˙s

Example Problem: Chain rule. #rvc‑er

Did you know?#rvc‑il

Gottfried Leibniz, one of the inventors of calculus, got the product rule wrong [Child, 1920, page 100; Cirillo, 2007]. In modern notation he computed the example ddx(x2+bx)(cx+d)=(2x+b)c and he stated that in general it was obvious that ddx(fg)=dfdxdgdx. He later realized his error and corrected it [Cupillari, 2004], but at least we know that product rules are tricky and not obvious, even for someone smart enough to invent calculus.

References

  • J. M. Child. The Early Mathematical Manuscripts of Leibniz. Open Court Publishing, 1920. (Google ebook, local copy).
  • M. Cirillo. Humanizing Calculus. The Mathematics Teacher, 101(1):23–27, 2007. (NCTM version, local copy)
  • A. Cupillari. Another look at the rules of differentiation. Primus: Problems, Resources, and Issues in Mathematics Undergraduate Studies, 14(3):193–200, 2004. DOI: 10.1080/10511970408984087.

Changing lengths and directions #rvc‑sd

Two useful derivatives are the rates of change of a vector's length and direction:

Derivative of vector length.#rvc‑el

˙a=˙aˆa

Derivation

Derivative of vector direction.#rvc‑eu

˙ˆa=1aComp(˙a,a)

Derivation

An immediate consequence of the derivative of direction formula is that the derivative of a unit vector is always orthogonal to the unit vector:

Derivative of unit vector is orthogonal.#rvc‑eu

˙ˆaˆa=0

Derivation

Recall that we can always write a vector as the product of its length and direction, so a=aˆa. This gives the following decomposition of the derivative of a.

Vector derivative decomposition.#rvc‑em

˙a=˙aˆaProj(˙a,a)+a˙ˆaComp(˙a,a)

Derivation

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Vector derivatives can be decomposed into length changes (projection onto a) and direction changes (complementary projection). Compare to Figure #rvv-fu. #rvc‑fm

Integrating vector functions #rvc‑si

The Riemann-sum definition of the vector integral is:

Vector integral.#rvc‑ei

t0a(τ)dτ=limNNi=1a(τi)ΔτSNτi=i1NΔτ=1N

In the above definition SN is the sum with N intervals, written here using the left-hand edge τi in each interval.

Time: t= 0 s
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Segments: N= 1

Integral of a vector function a(t), together with the approximation using a Riemann sum. #rvc‑fi

Just like vector derivatives, vector integrals only use the geometric concepts of scaling and addition, and do not rely on using a basis. If we do write a vector function in terms of a fixed basis, then we can integrate each component:

Vector integral in components.#rvc‑et

t0a(τ)dτ=(t0a1(τ)dτ)ˆı+(t0a2(τ)dτ)ˆȷ+(t0a3(τ)dτ)ˆk

Derivation

Warning: Integrating each component is only valid if the basis is fixed. #rvc‑wi

Example Problem: Integrating a vector function. #rvc‑xi

Warning: The dummy variable of integration must be different to the limit variable. #rvc‑wd